Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition, where $m\in [n,\infty )$ and $K\ge 0$ are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition and on compact manifolds equipped with $(?K,m)$-super Ricci flows.     

Counterexamples in calculus of variations in L∞ through the vectorial Eikonal equation

We show that, for any regular bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on ?Ω that solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the ∞-Laplace system arising in vectorial calculus of variations in $L^{\infty }$ does not suffice to characterise either limits of p-Harmonic maps as $p\to \infty$ or absolute minimisers in the sense of Aronsson.     

Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.     

On the Frobenius complexity of determinantal rings

We compute the Frobenius complexity for the determinantal ring of prime characteristic p obtained by modding out the $2\times 2$ minors of an $m\times n$ matrix of indeterminates, where $m>n?2$. We also show that, as $p\to \infty$, the Frobenius complexity approaches $m?1$.     

Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${?}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${?}_p$-norm.     

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the p−1 problem

We utilize the method of Bellman functions to derive new $L^p$-estimates of Littlewood–Paley type involving $p?1$. Among the applications to singular integrals we improve the $2(p?1)$ bounds for the Ahlfors–Beurling operator on $L^p(\mathbb{C})$ when $p\to \infty$. In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained. To cite this article: O. Dragi?evi?, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

Let $L=\mathrm{\Delta }-\mathrm{\nabla }\varphi \cdot \mathrm{\nabla }$ be a symmetric diffusion operator with an invariant measure $\mu (\mathrm{d}x)={\mathrm{e}}^{-\varphi (x)}\mathrm{d}x$ on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup $P_t={\mathrm{e}}^{tL}$ has the $C_0$-diffusion property and is unique in $L^1(M,\mu )$. As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the $L^1$-uniqueness of the intrinsic Schrödinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem.     

Existence of the global solution for the parabolic Monge–Ampère equations on compact Riemannian manifolds

On a compact Riemannian manifold $(M,g)$, we consider the existence and nonexistence of global solutions for the parabolic Monge–Ampère equation(?)$\{\begin{array}{c}\frac{?}{?t}\phi =\log \left(\frac{\det (g+\mathit{Hess}\phi )}{\det g}\right)?\lambda {\phi }^p?f(x),\hfill \\ \phi (x,0)={\phi }_0(x).\hfill \end{array}$ Here $p>1$ and λ are real parameters. $?f,{\phi }_0:M\to (0,+\infty )$ are smooth functions on M. If $\lambda >0$, then the solution φ of (?) exists for all times t and ${\phi }_t=\phi (?,t)$ converges exponentially towards a solution ${\phi }_{\infty }$ of its stationary equation as $t\to \infty$. In the case of $\lambda <0$, it does not have the global solution of (?). Thus we obtain the nonexistence of the positive solution for the stationary equation of (?).     

Some asymptotic properties of discrete means

In part 1, given n different ways of averaging n positive numbers, we iterate the resulting map in $(0,\infty {)}^n$. We prove convergence toward the diagonal, with rate estimates under smoothness assumptions. In part 2, we consider the elementary symmetric means of order p applied to the values $a_i=a(i/n),1⩽i⩽n$, of a given continuous positive function a on the normalized interval $[0,1]$ and we let $p=f(n)$. When ${\lim }_{n\to \infty }f(n)/n=0$, we prove that it admits a limit as $n\to \infty$, called the f-mean of a, which moreover coincides with ${\int }_0^{1}a(x)\mathrm{d}x$ whenever $f(n)=o(\log n)$. We record similar, quite immediate, results on the geometric side $p=n-f(n)$.